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Simplifying 2a(a + -4) + 15 = 13 Reorder the terms: 2a(-4 + a) + 15 = 13 (-4 * 2a + a * 2a) + 15 = 13 (-8a + 2a2) + 15 = 13 Reorder the terms: 15 + -8a + 2a2 = 13 Solving 15 + -8a + 2a2 = 13 Solving for variable 'a'. Reorder the terms: 15 + -13 + -8a + 2a2 = 13 + -13 Combine like terms: 15 + -13 = 2 2 + -8a + 2a2 = 13 + -13 Combine like terms: 13 + -13 = 0 2 + -8a + 2a2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(1 + -4a + a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + -4a + a2)' equal to zero and attempt to solve: Simplifying 1 + -4a + a2 = 0 Solving 1 + -4a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -4a + -1 + a2 = 0 + -1 Reorder the terms: 1 + -1 + -4a + a2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -4a + a2 = 0 + -1 -4a + a2 = 0 + -1 Combine like terms: 0 + -1 = -1 -4a + a2 = -1 The a term is -4a. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4a + 4 + a2 = -1 + 4 Reorder the terms: 4 + -4a + a2 = -1 + 4 Combine like terms: -1 + 4 = 3 4 + -4a + a2 = 3 Factor a perfect square on the left side: (a + -2)(a + -2) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (a + -2) equal to 1.732050808 and -1.732050808.Subproblem 1
a + -2 = 1.732050808 Simplifying a + -2 = 1.732050808 Reorder the terms: -2 + a = 1.732050808 Solving -2 + a = 1.732050808 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = 1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + a = 1.732050808 + 2 a = 1.732050808 + 2 Combine like terms: 1.732050808 + 2 = 3.732050808 a = 3.732050808 Simplifying a = 3.732050808Subproblem 2
a + -2 = -1.732050808 Simplifying a + -2 = -1.732050808 Reorder the terms: -2 + a = -1.732050808 Solving -2 + a = -1.732050808 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = -1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + a = -1.732050808 + 2 a = -1.732050808 + 2 Combine like terms: -1.732050808 + 2 = 0.267949192 a = 0.267949192 Simplifying a = 0.267949192Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.732050808, 0.267949192}Solution
a = {3.732050808, 0.267949192}
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